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Homemodule

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# module

Let $R$ be a ring with identity. A left module $M$ over $R$ is a set with two binary operations, $+:M\times M\longrightarrow M$ and $\cdot:R\times M\longrightarrow M$, such that

1. $(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$ for all $\mathbf{u},\mathbf{v},\mathbf{w}\in M$

2. $\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$ for all $\mathbf{u},\mathbf{v}\in M$

3. There exists an element $\mathbf{0}\in M$ such that $\mathbf{u}+\mathbf{0}=\mathbf{u}$ for all $\mathbf{u}\in M$

4. For any $\mathbf{u}\in M$, there exists an element $\mathbf{v}\in M$ such that $\mathbf{u}+\mathbf{v}=\mathbf{0}$

5. $a\cdot(b\cdot\mathbf{u})=(a\cdot b)\cdot\mathbf{u}$ for all $a,b\in R$ and $\mathbf{u}\in M$

6. $a\cdot(\mathbf{u}+\mathbf{v})=(a\cdot\mathbf{u})+(a\cdot\mathbf{v})$ for all $a\in R$ and $\mathbf{u},\mathbf{v}\in M$

7. $(a+b)\cdot\mathbf{u}=(a\cdot\mathbf{u})+(b\cdot\mathbf{u})$ for all $a,b\in R$ and $\mathbf{u}\in M$

A left module $M$ over $R$ is called *unitary* or *unital* if $1_{R}\cdot\mathbf{u}=\mathbf{u}$ for all $\mathbf{u}\in M$.

A (unitary or unital) *right module* is defined analogously, except that the function $\cdot$ goes from $M\times R$ to $M$ and the scalar multiplication operations act on the right. If $R$ is commutative, there is an equivalence of categories between the category of left $R$–modules and the category of right $R$–modules.

## Mathematics Subject Classification

13-00*no label found*16-00

*no label found*20-00

*no label found*44A20

*no label found*33E20

*no label found*30D15

*no label found*

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## Attached Articles

## Corrections

correction of statement by remag12 ✓

left module definition by remag12 ✓

correction of typographical error by remag12 ✘

correction of left module and right module definition of unital not unitary module by remag12 ✘

## Comments

## correction of definition

The module definition does not require an identity element in the ring. Those satisfying 1 * m = m for all m in the module are called

unitary modules.

-- S.A. G.

## correction of typographicalerror of unital module

Left (right) R modules M for rings R with identity 1 such that 1 * m = m (m * 1 = m) for all m in M are call unital modules.

-- S. A. G.

## typographical correction of unital module

Left (right) R modules M for rings R with identity 1 such that 1 * m = m (m * 1 = m) for all m in M are called unital modules.

-- S. A. G.

## simplifying the definition

Properties 1 through 4 are just the propreties of an abelian group. Wouldn't it be simpler to say that a module is an abelian group (M,+) with a binary operation . : R x M -> M that satisfies properties 5 through 7?